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CATEGORIES:Junior Algebra and Number Theory seminar
SUMMARY:Quotients of groups by torsion elements - Dr. Maur
ice Chiodo\, The University of Milan
DTSTART;TZID=Europe/London:20121123T140000
DTEND;TZID=Europe/London:20121123T150000
UID:TALK41242AThttp://talks.cam.ac.uk
URL:http://talks.cam.ac.uk/talk/index/41242
DESCRIPTION:It is well known that if we take a group G\, and q
uotient out by the normal closure of all non-trivi
al commutators\, then the resulting group is abeli
an (and in some sense “universal”). However\, if w
e instead quotient out by the normal closure of al
l torsion elements of G\, then the resulting group
need not be torsion-free. But\, by taking a count
ably infinite “tower” of quotients of torsion elem
ents\, we do eventually come to a torsion-free gro
up\, which is universal in the same way that the a
belianisation is.\nWe give an explicit constructio
n of a finitely presented group which is not torsi
on-free\, and for which the first quotient by tors
ion elements is again not torsion-free. We extend
this construction\,\nand combine it with some clas
sical embedding results\, to construct a finitely
generated (and recursively presented) group for wh
ich no finite iteration of quotients by torsion el
ements yields a\ntorsion-free group. This is a joi
nt work with Rishi Vyas.
LOCATION:MR4
CONTACT:Joanna Fawcett
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